Answers to Giancoli, section 24-5 questions 18-25
18. 0.0236 radians (or 1.35 degrees); 19. 952 nm; 20. 2.4 m; 21. 5.61 µm; 22. 8.46 cm; 23. 0.0230 mm; 24. 2.2 cm; 25.a) max b = lambda b) 400 nm
Note you can use the simulation to get some of the answers. Please discuss solutions in a discussion thread, NOT on this page. DJY 25/09
Here is the link to the single slit diffraction equation derivation that Matt and Elvis will explain again on these summary pages. Diffraction Derivation
4.5.1 Describe the reflection and transmission of waves at a boundary between two media
Contents:
A) One-Dimensional Reflection
B) Two-Dimensional Reflection
A) One-Dimensional waves at a boundary between two mediums
Reflection occurs when waves travels from one medium into another medium or hits an object.
Example of reflection: echo.
Two types of reflections that can occur in a string: fixed-end and free-end reflection.
1) Fixed End Reflection
from http://ddart.net/science/physics/physics_tutorial/Class/waves/u10l3a2.gif
This type of reflection produces 180° phase change. The reason for the inverted reflected pulse can be explained by Newton's Third Law. The string originally exerts an upward force on the support, which will exert a downward force on the string, causing it to create an inverted reflected pulse.
2) Free End Reflection
from http://ddart.net/science/physics/physics_tutorial/Class/waves/u10l3a4.gif
(Diagram from http://ddart.net/science/physics/physics_tutorial/Class/waves/u10l3a4.gif)
Free end reflection does not produce an inversion in the reflected pulse.The reflected pulse is not inverted, since there is no support to exert a downward force on the incident pulse. The cause for the reflected pulse can be attributed to the overshooting of the incident pulse, which causes an upward pull on the string.
B) Two-Dimensional Reflection
In a two dimensional reflection, there is an incident and a reflected angle.
Law of reflection:
Angle of incident = Angle of reflection (this is true when the incident angle is less than 90° to the normal).
(Diagram from http://library.thinkquest.org/26162/fig3-1.gif)
This diagram includes wave fronts and rays, and shows a wave reflecting off a straight barrier.
4.5.2 Refraction
Contents: A) Defintion
B) Example
C) Snell's Law
D) Refractive Indices
E) Critical Angles and Total Internal Reflection
A) Refraction: The bending of light as it passes from one medium to another medium. The bending occurs because when a wavefront travels from one medium to another, the wave speed changes. Examples include waves traveling from glass to air, air to water, deep water to shallow water (as shwon by ripples), and so forth.
In this diagram, wavefronts travel from deep to shallow water. As the wave travels to another medium, it bends; in this case, it bends towards the normal. The wave speed and wavelength decreases when it enters the shallow water, but the frequency stays the same. Wave velocity is determined by the medium, while frequency is determined by the source.
C) Snell's Law:
Snell's is given by the equation below.
or it can be rearranged as
Where θ1 is the angle of incidence, θ2 is the angle of refraction, and the refractive index (n) in the equation corresponds to the medium with the same subscript.
This diagram shows the refraction of a light ray passing from a fast medium (air) to a slower medium (glass).
Also, the light ray is partially reflected, as demonstrated by the reflected ray in air. In this case, that angle is equal to the incident angle.
Below is just an animation which I found that is related to Snell's Law, but involves wavefronts instead of a light rays. As can be seen, when the waves enter the bottom medium, the wavelengths are all reduced, meaning that the velocity is lower, and the object is denser.
(Animation from http://en.wikipedia.org/wiki/Law_of_refraction)
This animation shows the wavefronts from a point source. (Note that the region below the gray line has a higher refractive index than the top).
D) The Refractive Index:
The refractive index depends on the material of the medium.
Index of Refraction of light is the ratio of the speed of light in vacuum to the speed of light in that material. (http://theory.uwinnipeg.ca/physics/light/node5.html)
This is given by another equation.
c = speed of light = 3×10⁸m/s
v = velocity of light in the medium (m/s)
(Note: n≥1)
Commonly used refractive indices:
refractive index of air = 1.00
refractive index of water = 1.33
refractive index of glass = 1.52
refractive index of zircon = 1.92
refractive index of diamond = 2.42
SUMMARY
When light ray travels from a fast to slow medium, the refracted ray bends towards the normal. When a light ray travels from a slow to fast medium, it bends away from the normal.
E) Critical Angle and Total Internal Reflection Critical angle is the incident angle that produces a refracted angle of 90°
(Diagram from http://www.gcsescience.com/Critical-Angle.gif)
This shows the critical angle which lies along the medium boundary where the refracted angle is 90°.
Total internal reflection: when incident angle > critical angle, there is no refraction and light is reflected internally.
(Diagram from http://www.gcsescience.com/Total-Internal-Reflection.gif)
Note that incident angle = reflected angle.
An important note is that in order for there to be total internal reflection with light rays, the source must be emitted from the denser medium. The critical angle can be calculated with Critical Angle =arcsin(Refractive Index 2/Refractive Index 1). Refractive Index 1 is the refractive index of the medium in which the incident ray passes through. It is also the denser medium.
The greater the value of wavelength/slit width ratio, the more diffraction occurs.
Diffraction also depends on wavelength.
(Diagram from http://en.wikipedia.org/wiki/Diffraction)
The larger the wavelength, the greater the diffraction.
Also seen from the diagram above (if looked at carefully) are areas almost like rays coming from the aperture (destructive interference).
As for when there are obstacles, the amount of diffraction is also affected.
wavelength >> object width: bends around as if its not there (no shadow region)
wavelength > object width: bends around, but into shadow region
wavelength < object width: little bending into shadow region.
In diffraction with an aperature the opening acts as infinite point sources to generate new waves.
11.3 Diffraction
11.3.1 Sketch the variation with angle of diffraction of relative intensity of light diffracted at a single slit.
Diffraction helped demosntrate that light has wave characteristics, for in both the single- and double-slit experiments, the explanation for the light and dark fringes involved the concept of light diffracting and the interference of waves.
(Diagram from http://content.answers.com/main/content/wp/en/thumb/8/81/300px-Diffraction1.png)
This interference pattern can be observed when light is projected through a single slit and onto a screen. As the seen from the diagram there are areas of higher intensity light (constructive wave interference) and areas of no light (destructive wave interference).
For the single-slit diffration pattern, refer the hyperlink presented above in the previous section.
For the double-slit diffraction pattern, high intensity light occurs at areas where the light waves emitted from the two different slits constructively interfere. This occurs when the path difference of the light waves happen to have a path difference of n lambdas (n is an integer). Destructive interference occurs when the path difference of the light waves happen to have a path difference of (n+0.5) lambdas (n is an integer).
Applets:
The single and double slit simulations below help to visualize diffraction through a slit.
Single & Double Slit Simulation Java Applet:
18. 0.0236 radians (or 1.35 degrees); 19. 952 nm; 20. 2.4 m; 21. 5.61 µm; 22. 8.46 cm; 23. 0.0230 mm; 24. 2.2 cm; 25.a) max b = lambda b) 400 nm
Note you can use the simulation to get some of the answers. Please discuss solutions in a discussion thread, NOT on this page. DJY 25/09
Here is the link to the single slit diffraction equation derivation that Matt and Elvis will explain again on these summary pages.
Diffraction Derivation
4.5.1 Describe the reflection and transmission of waves at a boundary between two media
Contents:
A) One-Dimensional Reflection
B) Two-Dimensional Reflection
A) One-Dimensional waves at a boundary between two mediums
1) Fixed End Reflection
(Diagram from http://ddart.net/science/physics/physics_tutorial/Class/waves/u10l3a2.gif)
This type of reflection produces 180° phase change. The reason for the inverted reflected pulse can be explained by Newton's Third Law. The string originally exerts an upward force on the support, which will exert a downward force on the string, causing it to create an inverted reflected pulse.
2) Free End Reflection
(Diagram from http://ddart.net/science/physics/physics_tutorial/Class/waves/u10l3a4.gif)
Free end reflection does not produce an inversion in the reflected pulse.The reflected pulse is not inverted, since there is no support to exert a downward force on the incident pulse. The cause for the reflected pulse can be attributed to the overshooting of the incident pulse, which causes an upward pull on the string.
B) Two-Dimensional Reflection
In a two dimensional reflection, there is an incident and a reflected angle.
Law of reflection:
Angle of incident = Angle of reflection (this is true when the incident angle is less than 90° to the normal).
(Diagram from http://library.thinkquest.org/26162/fig3-1.gif)
Wave reflection
(Diagram from http://www.gcsescience.com/Reflection-Water-Waves.gif)
This diagram includes wave fronts and rays, and shows a wave reflecting off a straight barrier.
4.5.2 Refraction
Contents:
A) Defintion
B) Example
C) Snell's Law
D) Refractive Indices
E) Critical Angles and Total Internal Reflection
A) Refraction: The bending of light as it passes from one medium to another medium. The bending occurs because when a wavefront travels from one medium to another, the wave speed changes. Examples include waves traveling from glass to air, air to water, deep water to shallow water (as shwon by ripples), and so forth.
B) Example: Ripple tank
A ripple tank is a shallow glass water tank used to demonstrate refraction.
(Diagram from http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/waves/u10l3b.html)
In this diagram, wavefronts travel from deep to shallow water. As the wave travels to another medium, it bends; in this case, it bends towards the normal. The wave speed and wavelength decreases when it enters the shallow water, but the frequency stays the same. Wave velocity is determined by the medium, while frequency is determined by the source.
C) Snell's Law:
Snell's is given by the equation below.
or it can be rearranged as
Where
θ1 is the angle of incidence,
θ2 is the angle of refraction,
and the refractive index (n) in the equation corresponds to the medium with the same subscript.
Snell's law defines the relationship between the incident and refracted angle as a light ray passes between media.
(Diagram from http://upload.wikimedia.org/wikipedia/commons/6/62/Example_snells_law.gif)
This diagram shows the refraction of a light ray passing from a fast medium (air) to a slower medium (glass).
Also, the light ray is partially reflected, as demonstrated by the reflected ray in air. In this case, that angle is equal to the incident angle.
Below is just an animation which I found that is related to Snell's Law, but involves wavefronts instead of a light rays. As can be seen, when the waves enter the bottom medium, the wavelengths are all reduced, meaning that the velocity is lower, and the object is denser.
(Animation from http://en.wikipedia.org/wiki/Law_of_refraction)
This animation shows the wavefronts from a point source. (Note that the region below the gray line has a higher refractive index than the top).
D) The Refractive Index:
The refractive index depends on the material of the medium.
Index of Refraction of light is the ratio of the speed of light in vacuum to the speed of light in that material. (http://theory.uwinnipeg.ca/physics/light/node5.html)
This is given by another equation.
c = speed of light = 3×10⁸m/s
v = velocity of light in the medium (m/s)
(Note: n≥1)
Commonly used refractive indices:
refractive index of air = 1.00
refractive index of water = 1.33
refractive index of glass = 1.52
refractive index of zircon = 1.92
refractive index of diamond = 2.42
SUMMARY
When light ray travels from a fast to slow medium, the refracted ray bends towards the normal. When a light ray travels from a slow to fast medium, it bends away from the normal.
E) Critical Angle and Total Internal Reflection
Critical angle is the incident angle that produces a refracted angle of 90°
(Diagram from http://www.gcsescience.com/Critical-Angle.gif)
This shows the critical angle which lies along the medium boundary where the refracted angle is 90°.
Total internal reflection: when incident angle > critical angle, there is no refraction and light is reflected internally.
(Diagram from http://www.gcsescience.com/Total-Internal-Reflection.gif)
Note that incident angle = reflected angle.
An important note is that in order for there to be total internal reflection with light rays, the source must be emitted from the denser medium. The critical angle can be calculated with Critical Angle =arcsin(Refractive Index 2/Refractive Index 1). Refractive Index 1 is the refractive index of the medium in which the incident ray passes through. It is also the denser medium.
4.5.3-4 Diffraction
Definition of diffraction: the phenomena of waves when obstructing an obstacle or passing through apertures.
(http://www.saburchill.com/physics/chapters2/0008.html)
Factors that affect the amount of diffraction that occurs:
Below is a diagram of a diffraction through different aperture sizes.
(Diagrams from http://www.gcsescience.com/Diffraction-Water-Waves.gif and http://www.gcsescience.com/No-Diffraction-Water-Waves.gif)
It can be seen that the smaller the aperture, the greater the diffraction.
The greater the value of wavelength/slit width ratio, the more diffraction occurs.
Diffraction also depends on wavelength.
(Diagram from http://en.wikipedia.org/wiki/Diffraction)
The larger the wavelength, the greater the diffraction.
Also seen from the diagram above (if looked at carefully) are areas almost like rays coming from the aperture (destructive interference).
As for when there are obstacles, the amount of diffraction is also affected.
In diffraction with an aperature the opening acts as infinite point sources to generate new waves.
11.3 Diffraction
11.3.1 Sketch the variation with angle of diffraction of relative intensity of light diffracted at a single slit.
Diffraction helped demosntrate that light has wave characteristics, for in both the single- and double-slit experiments, the explanation for the light and dark fringes involved the concept of light diffracting and the interference of waves.
(Diagram from http://content.answers.com/main/content/wp/en/thumb/8/81/300px-Diffraction1.png)
This interference pattern can be observed when light is projected through a single slit and onto a screen. As the seen from the diagram there are areas of higher intensity light (constructive wave interference) and areas of no light (destructive wave interference).
For the single-slit diffration pattern, refer the hyperlink presented above in the previous section.
For the double-slit diffraction pattern, high intensity light occurs at areas where the light waves emitted from the two different slits constructively interfere. This occurs when the path difference of the light waves happen to have a path difference of n lambdas (n is an integer). Destructive interference occurs when the path difference of the light waves happen to have a path difference of (n+0.5) lambdas (n is an integer).
Applets:
The single and double slit simulations below help to visualize diffraction through a slit.
Single & Double Slit Simulation Java Applet:
simulations from: http://www.walter-fendt.de/ph14e/index.html
By Elvis and Matt
Edited and Commented by Jisoo and Jeffrey